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**Harvard**

Tykesson, J. (2008) *Continuum Percolation in non-Euclidean Spaces*. Göteborg : Chalmers University of Technology (Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie, nr: 2776).

** BibTeX **

@book{

Tykesson2008,

author={Tykesson, Johan},

title={Continuum Percolation in non-Euclidean Spaces},

isbn={978-91-7385-095-7},

abstract={In this thesis we first consider the Poisson Boolean model of
continuum percolation in $n$-dimensional hyperbolic space ${\mathbb
H}^n$. Let $R$ be the radius of the balls in the model, and
$\lambda$ the intensity of the underlying Poisson process. We show
that if $R$ is large enough, then there is an interval of intensities
such that there are infinitely many unbounded components in the
covered region. For $n=2$, more refined results are obtained.
We then consider the model on some more general spaces. For a large
class of homogeneous spaces, it is established that if $\lambda$ is such that
there is a.s.\ a unique unbounded component in the covered region,
then this is also the case for any $\lambda_1>\lambda$. In ${\mathbb
H}^2\times{\mathbb R}$ it is proved that if $\lambda$ is critical
for a.s.\ having a unique unbounded component in the covered region,
then there is a.s.\ not a unique unbounded component.
Finally, we consider another aspect of continuum percolation in
${\mathbb H}^2$. We show that in the Poisson Boolean model, there are
intensities for which infinite geodesics are contained in unbounded
components of the covered region. This is also shown for the vacant
region, as well as for a larger class of continuum percolation models.
We also consider some dynamical models.},

publisher={Institutionen för matematiska vetenskaper, Chalmers tekniska högskola,},

place={Göteborg},

year={2008},

series={Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie, no: 2776},

keywords={Bernoulli percolation, continuum percolation, dependent percolation, double phase transition, hyperbolic space, Poisson Boolean model, geodesic percolation},

note={96},

}

** RefWorks **

RT Dissertation/Thesis

SR Electronic

ID 69483

A1 Tykesson, Johan

T1 Continuum Percolation in non-Euclidean Spaces

YR 2008

SN 978-91-7385-095-7

AB In this thesis we first consider the Poisson Boolean model of
continuum percolation in $n$-dimensional hyperbolic space ${\mathbb
H}^n$. Let $R$ be the radius of the balls in the model, and
$\lambda$ the intensity of the underlying Poisson process. We show
that if $R$ is large enough, then there is an interval of intensities
such that there are infinitely many unbounded components in the
covered region. For $n=2$, more refined results are obtained.
We then consider the model on some more general spaces. For a large
class of homogeneous spaces, it is established that if $\lambda$ is such that
there is a.s.\ a unique unbounded component in the covered region,
then this is also the case for any $\lambda_1>\lambda$. In ${\mathbb
H}^2\times{\mathbb R}$ it is proved that if $\lambda$ is critical
for a.s.\ having a unique unbounded component in the covered region,
then there is a.s.\ not a unique unbounded component.
Finally, we consider another aspect of continuum percolation in
${\mathbb H}^2$. We show that in the Poisson Boolean model, there are
intensities for which infinite geodesics are contained in unbounded
components of the covered region. This is also shown for the vacant
region, as well as for a larger class of continuum percolation models.
We also consider some dynamical models.

PB Institutionen för matematiska vetenskaper, Chalmers tekniska högskola,

T3 Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie, no: 2776

LA eng

LK http://www.math.chalmers.se/Stat/Research/Preprints/Doctoral/2008/2.pdf

OL 30